Editing Superalgebra (section)

==Formal definition==

Let ''K'' be a [[commutative ring]]. In most applications, ''K'' is a [[field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] 0, such as '''R''' or '''C'''.

A '''superalgebra''' over ''K'' is a [[module (mathematics)|''K''-module]] ''A'' with a [[direct sum of modules|direct sum]] decomposition
:<math>A = A_0\oplus A_1</math>
together with a [[bilinear map|bilinear]] multiplication ''A'' &times; ''A'' &rarr; ''A'' such that
:<math>A_iA_j \sube A_{i+j}</math>
where the subscripts are read [[Modular arithmetic|modulo]] 2, i.e. they are thought of as elements of '''Z'''<sub>2</sub>.

A '''superring''', or '''Z'''<sub>2</sub>-[[graded ring]], is a superalgebra over the ring of [[integer]]s '''Z'''.

The elements of each of the ''A''<sub>''i''</sub> are said to be '''homogeneous'''. The '''parity''' of a homogeneous element ''x'', denoted by {{abs|''x''}}, is 0 or 1 according to whether it is in ''A''<sub>0</sub> or ''A''<sub>1</sub>. Elements of parity 0 are said to be '''even''' and those of parity 1 to be '''odd'''. If ''x'' and ''y'' are both homogeneous then so is the product ''xy'' and <math>|xy| = |x| + |y|</math>.

An '''associative superalgebra''' is one whose multiplication is [[associative]] and a '''unital superalgebra''' is one with a multiplicative [[identity element]]. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.

A '''[[commutative superalgebra]]''' (or supercommutative algebra) is one which satisfies a graded version of [[commutativity]]. Specifically, ''A'' is commutative if

:<math>yx = (-1)^{|x||y|}xy\,</math>

for all homogeneous elements ''x'' and ''y'' of ''A''. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called ''supercommutative'' in order to avoid confusion.<ref>{{harvnb|Varadarajan|2004|p=87}}</ref>